# Definition:Increasing/Mapping

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## Definition

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is **increasing** if and only if:

- $\forall x, y \in S: x \preceq_1 y \ \implies \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Note that this definition also holds if $S = T$.

## Also known as

An **increasing** mapping is also known as **order-preserving**, **isotone** and **non-decreasing**.

Some authors refer to this concept as a **monotone mapping**, but that term has a different meaning on ProofWiki.

## Also defined as

Some sources insist at the point of definition that $\phi$ be an injection for it to be definable as **order-preserving**, but this is conceptually unnecessary.

## Also see

- Definition:Strictly Increasing Mapping
- Definition:Decreasing Mapping
- Definition:Monotone Mapping
- Results about
**increasing mappings**can be found here.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Orderings - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$ - 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S \text I.2$ - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$